(b) the linear system \(|D|\)  contains an irreducible smooth curve \(\iff \)  it contains an irreducible curve
\(\iff a=0, b=1\)  or \(a=1, b=0\)  or \(a>0, b>an\)  or \(a>0, b=an, n>0\) .
The Hirzebruch surface \(\mathbb {F}_0\)  is isomorphic to \(\mathbb {P}^1 \times \mathbb {P}^1\) , so \(\mathrm {Pic}\,(\mathbb {F}_0)\)  is generated by two lines belonging to distinct rulings. But also in this case, we denote these generators by \( C_0 \)  and \(L\)  (like in the case of \(\mathbb {F}_n\) , where \(n >0\) ), even if their roles can be exchanged.
Theorem 4.6 
Let \(X \sim \gamma C_0 + \lambda L\)  be an irreducible smooth curve on \(\mathbb {F}_n\)  with \(\gamma \ge 2\)  and not isomorphic to a plane curve.
Dividing \(\lambda -n-1\)  by \(\gamma -2\) , let us denote by \(\beta \)  the quotient and \(\epsilon \)  the remainder, i.e.
\(\beta := \left[ \frac{\lambda -n-1}{\gamma -2}\right] =\frac{\lambda -n-1-\epsilon }{\gamma -2}, \quad 0 \le \epsilon \le \gamma -3.\) 
Moreover, set
\(r:=2\beta +1-n \quad \hbox{and} \quad d:=\gamma (\beta - n) + \lambda .\) 
Consider the complete linear system \(|H|\)  on \(\mathbb {F}_n\)  given by
\(H \sim C_0 + \beta L.\) 
Then
\(i)\)
\(\beta > n\) , for all \(n \ge 0\) ;
\(ii)\)
the morphism \(\varphi _H\)  embeds \(\mathbb {F}_n\)  in \(\mathbb {P}^r\) ;
\(iii)\)
\(\varphi _H(X)\)  is a curve of degree \(d\) .

Assume in addition that
\(\lambda \ge \frac{\gamma (\gamma +n -2)}{2}.\) 
Then
\(iv)\)
\(m_{d,r} = \gamma -1\)  and \( \epsilon _{d,r} = \epsilon \) ;
\(v)\)
\(\varphi _H(X)\)  is an extremal curve in \(\mathbb {P}^r\) .


Note first that, by
assumption, \(X\)  is irreducible and smooth. Then, by Proposition REF  and the assumption \(\gamma \ge 2\) , we have
\(\lambda > 0, \quad \hbox{if} \quad n =0\) 
and
\(\lambda \ge \gamma n, \quad \hbox{if} \quad n >0\) 
\(i)\) 
If \(n =0\) , it is clear that \(\beta \ge 1 \iff \lambda -1 \ge \gamma -2\) . If this is not the case, we observe that we can change the role of \(\gamma \)  and \(\lambda \) , since on \(\mathbb {F}_0 \cong \mathbb {P}^1 \times \mathbb {P}^1\)  we can choose arbitrarily one of the two rulings.
If \(n=1\) , then \(\beta > 1\)  if and only if \(\lambda -2 > \gamma -2\)  and this holds by the assumption that \(X\)  is not isomorphic to a plane curve.
Indeed, in general, any irreducible curve \(\gamma C_0 + \lambda L\) 
on \(\mathbb {F}_1\)  satisfies \(\lambda \ge \gamma \)  by (REF ). In particular, if \(\lambda =\gamma \) , we have \(\beta =1\)  and \(H = C_0 + L\) . On one hand, it is clear that the linear system \(|H|\)  does not induce an embedding of \(\mathbb {F}_1\) , as it maps surjectively to \(\mathbb {P}^2\)  and corresponds to the contraction of the exceptional curve \(C_0\) ; it is well known that this is the blowing up morphism
\(\mathbb {F}_1 \rightarrow \mathbb {P}^2\)  of the plane in a point.
On the other hand, under such a morphism, any smooth irreducible curve \(X\sim \gamma (C_0 + L)\)  is mapped isomorphically to a smooth degree \(\gamma \)  plane curve, which contradicts our assumption.
If \(n>1\) , obviously \(\beta > n\)  if and only if
\(\lambda -n-1 > n(\gamma -2).\) 
i.e. \(\lambda > n(\gamma -1) +1\) .
But this holds since \(n > 1\)  implies \(n\gamma -n+1 < n\gamma \)  and, by (REF ), \(n\gamma \le \lambda \) .
\(ii)\) 
Again by Proposition REF , the linear system \(|H| = | C_0 + \beta L|\)  is very ample if and only if \(\beta > 0\)  if \(n =0\)  or \(\beta > n\)  if \(n >0\)  and this holds by \((i)\) . Hence \(\varphi _H\)  is an embedding.
Moreover, we have the well–known formula (see, for instance [1]}):
\(r+1 = h^0( \mathcal {O}_{\mathbb {F}_n} (C_0 + \beta L)) =2\beta +2 -n.\) 
\(iii)\) 
Therefore \(\varphi _H\)  embeds \(X\)  in \(\mathbb {P}^r\)  as a curve of degree
\((\gamma C_0+ \lambda L) \cdot (C_0 + \beta L)=\gamma (\beta - n) + \lambda = d,\) 
as required.
\(iv)\) 
In order to show that
\(m_{d,r} = \gamma -1\)  and \( \epsilon _{d,r} = \epsilon \) , it is enough to prove that
\(d -1 -( \gamma -1)(r-1) = \epsilon \quad \hbox{and} \quad 0 \le \epsilon \le r-2.\) 
Clearly
\(d -1 -( \gamma -1)(r-1) = \gamma (\beta - n) + \lambda -1 - ( \gamma -1)(2\beta -n ) = \lambda -n-1-\beta (\gamma -2)\) 
and, substituting the value of \(\beta \) , we obtain the requested equality.
In order to show that \(\epsilon \le r-2\) , note first that, for any \(n \ge 0\) , we have
\(\epsilon \le r-2 \iff \lambda -n-1-\beta (\gamma -2) \le 2\beta -n -1 \iff \lambda \le \beta \gamma .\) 
Since \(\epsilon \le \gamma -3\) , clearly
\(\beta =\frac{\lambda -n-1-\epsilon }{\gamma -2} \ge \frac{\lambda -n +2 - \gamma }{\gamma -2},\) 
so in order to show that \(\lambda \le \beta \gamma \) , it is enough to prove that
\(\lambda \le \gamma \frac{\lambda -n +2 - \gamma }{\gamma -2}\quad \iff \quad 2 \lambda \ge \gamma (\gamma +n -2)\) 
and this holds by assumption.
\(v)\) 
In order to prove that \(\varphi _H(X) \subset \mathbb {P}^r\)  is an extremal curve, we compute the genus \(g\)  of \(X\) 
using the Adjunction Formula obtaining
\(\begin{array}{ll}2g-2& =(K_{\mathbb {F}_n} +X) \cdot X =(-2C_0 -(2+n)L + \gamma C_0+\lambda L) \cdot (\gamma C_0+\lambda L)=\\&= 2(\lambda \gamma - \lambda -\gamma ) - n \gamma (\gamma -1),\\\end{array}\) 
which yields
\(g=\lambda \gamma - \lambda -\gamma +1 - \frac{n}{2} \gamma (\gamma -1) = (\lambda -1)(\gamma -1) - \frac{n}{2} \gamma (\gamma -1).\) 
Now we can compute the Castelnuovo bound
\(\pi (d,r) = m_{d,r}\left( \frac{m_{d,r}-1}{2} (r-1)+\epsilon _{d,r} \right) =(\gamma -1)\left( \frac{\gamma -2}{2} (2\beta -n)+\epsilon \right).\) 
Since
\(2\beta -n = 2 \; \frac{\lambda -n-1-\epsilon }{\gamma -2} - n = \frac{2\lambda -2-2\epsilon - n \gamma }{\gamma -2}\) 
we obtain
\(\pi (d,r) = (\gamma -1)\left( \frac{1}{2} (2\lambda -2-2\epsilon - n \gamma )+\epsilon \right) = (\gamma -1)\left(\lambda -1-\epsilon -n\gamma /2 + \epsilon \right)\) 
and, finally,
\(\pi (d,r) =(\gamma -1)(\lambda -1 -n\gamma /2 ) = (\gamma -1)(\lambda -1) - \frac{n}{2} \gamma (\gamma -1).\) 
Comparing this formula with (REF ), we see that \(\pi (d,r) = g\)  and hence \(\varphi _H(X)\)  is an extremal curve in \(\mathbb {P}^r\) .
Remark 4.7 By the irreducibility of \(X\) , we have
\(\lambda \ge \gamma n\)  from Proposition REF . As a consequence, the additional assumption in Theorem  REF
\(\lambda \ge \frac{\gamma (\gamma +n -2)}{2}.\) 
holds if \(n \ge \gamma -2\) .

In the sequel we will need to relate the gonality of \(X\)  with its \(m\) -ratio. Since Theorem REF  claims that the extremal curves, not isomorphic to plane curves, lie on a rational normal ruled surface, we here recall the following result of Martens (see [2]}) which describes such a relationship in the wider case of curves on ruled surfaces (see Definition REF ).
Theorem 4.8 
Let \(X\)  be a reduced and irreducible curve on a Hirzebruch surface \(\mathbb {F}_n\)  and assume that \(X\)  is not a fibre. Then the gonality of \(X\)  is computed by a ruling of the surface, unless \(n=1\)  and \(X \sim \alpha (C_0 + L)\)  for some \(\alpha \ge 2\) , in which case \(X\)  is isomorphic to a plane curve of degree \(\alpha \)  and its gonality is \(\alpha -1\) .

Remark 4.9 The exceptional case in Theorem REF  concerns curves of the type
\(X \sim \alpha (C_0 + L)\) 
on \(\mathbb {F}_1\) . We observe that such a situation never occurs in the framework of extremal curves of type (ii) and (iii) in Theorem 4.1.
Note first that for a rational ruled surface \(R \subset \mathbb {P}^r\)  of degree \(r-1\)  the hyperplane divisor \(H\)  satisfies
\(H \sim C_0 + \frac{r-1- C_0^2 }{ 2} L\)  by Remark REF . Hence
\(R \cong \mathbb {F}_1\)  if and only if
\(H \sim C_0 + \frac{r}{ 2} L.\) 
It follows that, in case (ii), we have
\(X \sim mH+L= m C_0 + \left(\frac{mr}{ 2} +1\right)L\) 
and, so, \(X \sim \alpha C_0 + \alpha L\)  if and only if \(\alpha =m= \frac{2}{ 2-r}\) , which is not possible for \(r \ge 2\) .
In case (iii) we have
\(X \sim (m+1)H-(r-2-\epsilon )L= (m+1) C_0 +\left(\frac{(m+1)r}{ 2}-r+2+\epsilon \right)L.\) 
hence \(X \sim \alpha C_0 + \alpha L\)  if and only if \(\alpha =m+1\)  and
\(m+1 = \frac{(m+1)r}{ 2}-r+2+\epsilon \quad \Rightarrow \quad \epsilon = (2-r) (m-1)/2.\) 
But \(\epsilon \ge 0\)  so we get a contradiction for \(r \ge 3\) .

The two results recalled above (Theorems REF  and REF )
lead to the following consequence, whose formulas
immediately come from (REF ) and (REF ).
Corollary 4.10 
Let \(X\)  be a \(\gamma \) -gonal extremal curve in \(\mathbb {P}^r\)  (where \(r \ge 3\) ) of degree \(d\) , genus \(g\)  and \(m\) -ratio \(m\) . If \(X\)  is not isomorphic to a plane curve, then there exists a rational normal ruled surface \(R\)  such that \(X \subset R \subset \mathbb {P}^r\)  and, setting \(\mathrm {Pic}\,(R) = \mathbb {Z}[H] \oplus \mathbb {Z}[L]\) , either:
i)
if \(X \in |mH+L|\)  on \(R\)  then \(\epsilon =0\)  and
\(m= \gamma \) .
Consequently,
\(d= \gamma (r-1) +1\) 
\(g= \frac{\gamma (\gamma -1)(r-1)}{2} .\) 
or
ii)
if \(X \in |(m+1)H- (r-\epsilon -2)L|\)  on \(R\)  then
\(m= \gamma -1\) .
Consequently,
\(d= (\gamma -1)(r-1)+\epsilon +1\) 
\(g=(\gamma -1) \left[\frac{\gamma -2}{2} \, (r-1) + \epsilon \right];\) 

In particular, the gonal series \(g^1_\gamma \)  on \(X\)  comes from the restriction of the fibration \(\pi : R \rightarrow \mathbb {P}^1\)  given by the ruling.

Remark 4.11 
Assume \(\gamma \ge 4\)  in Corollary REF  and consider the case \((ii)\) : \(m =\gamma -1\) .
If \(\epsilon = 0\)  and \(d \ge 3r -1\)  then, from (REF ), we have
\(d = ( \gamma -1)(r-1) + 1 \ge 3r -1\quad \Longrightarrow \quad (\gamma -4)r \ge \gamma -3 \quad \Longrightarrow \quad \gamma \ge 5.\) 

We shall need the following result (see [3]}).
Theorem 4.12 
Let \(X\)  be an extremal curve of degree \(d \ge 3r - 1\)  in \(\mathbb {P}^r\) . Then \(d_{r -1} = d - 1\)  and \(X\)  does not satisfy all slope inequalities.

Now we can state the main result of this section, which gives a negative answer to the question posed by Kato and Martens in [4]}, i.e. if the \(r\) -th slope inequality is violated for extremal curves in \(\mathbb {P}^r\)  for any \(r \ge 4\) .
Theorem 4.13 
Let \(X\)  be a \(\gamma \) -gonal extremal curve of degree \(d \ge 3r - 1\)  in \(\mathbb {P}^r\)  (\(r \ge 3\) ) where \(\gamma \ge 4\) . Then \(d_r=d\) .
If, in addition, \(X\)  is not isomorphic to a plane curve,
then \(X\)  satisfies the following conditions:
\(i)\)
\(d_{r+1}\le d+\gamma -1\) ;
\(ii)\)
by assuming one of the following sets of hypotheses:
\((a)\)
either \(\epsilon =0\) , \(m = \gamma \)  and \(r \ge \gamma -1\) ,
\((b)\)
or \(\epsilon \ge \gamma -2\) ,

then the \(r\) -th slope inequality holds, i.e.
\(\frac{d_r}{r} \ge \frac{d_{r+1} }{r+1}.\) 


From Theorem REF , we obtain
\(d_{r-1}= d-1\) . Since \(d_r > d_{r-1}= d-1\) , we get that \(d_r \ge d\) . On the other hand \(X\)  possesses a \(g^r_d\)  by
assumption, so \(d_r \le d\) , hence the first statement is proved.
\((i)\)  Since \(X \subset \mathbb {P}^r\)  is a curve of degree \(d\) , denoting by \(|H|\)  the hyperplane linear system in
\(\mathbb {P}^r\) , the induced linear system \(|H_X|\)  on \(X\)  is a linear series \(g^r_d\) .
From Corollary REF ,
\(X\)  is a \(\gamma \) –secant curve on a rational ruled surface \(R\)  and the gonal series \(g^1_\gamma \)  on \(X\)  comes from the restriction of \(\pi : R \rightarrow \mathbb {P}^1\) . So, for any \(P \in X\) , we set
\(\Gamma _P\)  to be the gonal divisor contining \(P\) , i.e. \(\Gamma _P = \pi ^{-1}(\pi (P)) \cap X\) .
Also observe that the general hyperplane \(H\)  cuts on \(R\)  an irreducible unisecant curve, say \(U_H\) . In particular, the general hyperplane \(H\)  does not contain any line of the ruling. Therefore for a general \(H_X \in g^r_d\)  and for any \(P \in H_X\) , we have
\(H_X \cap \Gamma _P = H \cap X \cap \Gamma _P = (H \cap R)\cap X \cap \Gamma _P = U_H \cap X \cap \Gamma _P= \lbrace P\rbrace .\) 
Setting \(\Gamma _P = P + Q_1 + \cdots + Q_{\gamma -1}\) , let us consider the divisor obtained by adding to \(H_X\)  the \(\gamma -1\)  further points of the gonal divisor, i.e. \(D = H_X + Q_1 + \cdots + Q_{\gamma -1}\) ;
we have
\(\deg D = \deg H_X +\gamma -1 = d+ \gamma -1.\) 
Now let us consider the canonical model \(X_K \subset \mathbb {P}^{g-1}\) ; here we can apply the Geometric Riemann-Roch Theorem to all the divisors \(\Gamma _P\) , \(H_X\)  and \(D\) . First we obtain
\(\dim \langle \Gamma _P \rangle = \deg \Gamma _P - h^0({\mathcal {O}}(\Gamma _P)) = \gamma -2\) 
and
\(\dim \langle H_X \rangle = \deg H_X - h^0({\mathcal {O}}(H_X)) = d-r-1.\) 
Consequently, since the intersection \( \langle \Gamma _P \rangle \cap \langle H_X \rangle \)  contains \(P\) , we have
\(\dim \langle D \rangle \le \dim \langle H_X \rangle +\gamma -2 = d-r+\gamma -3.\) 
Hence, again from the Geometric Riemann-Roch Theorem, we get that
\(t+1 := h^0({\mathcal {O}}(D)) = \deg D - \dim \langle D \rangle \ge d+\gamma -1 -(d-r+\gamma -3) = r+2.\) 
Therefore there exists an integer \(t \ge r+1\)  such that \(|D|\)  is of the form \(g^t_{d+\gamma -1}\) .
This implies that \(d_t \le d+\gamma -1\) . From
Lemma REF , we finally obtain that \(d_{r+1} \le d_t \le d+\gamma -1\) .
\((ii)\)  From the fact that \(d_r=d\)  and \((i)\) , it is enough to show that
\(\frac{d}{r} \ge \frac{d+\gamma -1}{r+1},\) 
or, equivalently,
\(d \ge r(\gamma -1).\) 
\((a)\)  Assume \(\epsilon =0\) , \(m = \gamma \)  and \(r +1 \ge \gamma \) . Then, using (REF ),
\(d= \gamma (r-1) +1 = r \gamma -\gamma +1 \ge r \gamma -(r+1) +1 = r(\gamma -1)\) 
i.e. (REF ), as required.
\((b)\) 
Assume \(\epsilon \ge \gamma -2 \ge 1\) . Then we express the degree using (REF ), obtaining
\(d= (\gamma -1)(r-1)+\epsilon +1 \ge (\gamma -1)(r-1)+\gamma -1 = r(\gamma -1)\) 
so we get (REF ), as required.
Let us observe that the condition that \(X\)  is not isomorphic to a plane curve is necessary since, otherwise, the inequality \((i)\)  does not hold (see
Remark REF ) and, even assuming \((b)\) , the \(r\) -th slope inequality fails.
Corollary 4.14
Let \(X\)  be a \(\gamma \) -gonal extremal curve in \(\mathbb {P}^r\) , where \(\gamma \ge 4\) .
If the degree \(d\)  satisfies
\(r(\gamma -1) \le d \le \gamma (r-1)+1\) 
then the \(r\) -th slope inequality holds.

Let us note first that, under the assumption (REF ), the curve \(X\)  cannot be isomorphic to a plane curve. If so, by Theorem REF , (i), we would have \(r=5\) , \(d=2k\)  and \(\gamma =k-1\)  by Max Noether Theorem, contradicting (REF ).
Now observe that the assumption implies
\(r(\gamma -1) \le \gamma (r-1)+1\quad \Rightarrow \quad r \ge \gamma -1.\) 
Since \(X\)  is an extremal curve and \(X\)  is not isomorphic to a plane curve, by Corollary REF , we have two possible cases.
In the first one, \(m=\gamma \) , \(\epsilon =0\)  and
\(d= \gamma (r-1) +1\)  from (REF ). Since \(r \ge \gamma - 1\) , by Theorem REF  we obtain that the \(r\) -th slope inequality holds.
In the second case, \(m=\gamma -1\)  and \(d= (\gamma -1)(r-1)+\epsilon +1\)  from (REF ). The assumption (REF ) yields
\((\gamma -1)(r-1)+\epsilon +1 \ge r(\gamma -1)\quad \Rightarrow \quad \epsilon \ge \gamma -2.\) 
Again by Theorem REF , we obtain that the \(r\) -th slope inequality holds.
The study above and, in particular, Corollary REF , can be summarized in the following table where we shall consider only curves which are not isomorphic to plane curves.
The first column concerns the increasing degree and the last one the \(r\) -th slope inequality.
In particular, the case \(d=2r\) , here omitted, has been described in
Remark REF .
The first four lines of the table will be considered in detail in the next section.

<TABLE>
Table 1
Violating cases
This section concerns the first four lines of the Table 1. More precisely, extremal curves of degree in the initial range are forced to have certain gonality (see Proposition REF ). Moreover, the \(r\) -th slope inequality is violated by extremal curves of degree \(d=3r-1\)  (see Proposition REF ). Finally, we will treat the case of curves of degree \(d=3r-2\)  in Proposition REF  and Theorem REF .
Proposition 5.1 
Let \(X \subset \mathbb {P}^r\)  (where \(r \ge 3\) ) be an extremal curve of degree \(d\)  and genus \(g\) , not isomorphic to a plane curve and such that
\(2r+1 \le d \le 3r -1.\) 
Then:
i) if \(2r+1 \le d \le 3r -3\)  then \(X\)  is trigonal (\(m=2\) , \(\epsilon \ge 2\) );

ii) if \(d = 3r -2\)  then \(X\)  is either trigonal or fourgonal (where \(m=3\) , \(\epsilon =0\) );

iii) if \(d = 3r -1\)  then \(X\)  is fourgonal (where \(m=3\) , \(\epsilon =1\) ).
Let us recall that \(d = m(r-1) + \epsilon +1\)  where \(m =[(d-1) / (r-1)]\) .
\((i)\)  In this case
\(\frac{d-1}{r-1} \le \frac{3r-4}{r-1} <3\) 
hence \(m=2\) . So \(d = 2(r-1) + \epsilon +1\)  and the bound \(d \ge 2r +1\)  implies \(\epsilon \ge 2\) . Therefore we are in the case of Corollary REF  - (ii), hence \(X\)  admits a \(g^1_3\) , i.e. it is a trigonal curve.
\((ii)\)  In this case
\(\frac{d-1}{r-1} = \frac{3r-3}{r-1} =3\) 
hence \(m=3\) . Therefore \(d = 3(r-1) + \epsilon +1 = 3r-2\)  and, so, \(\epsilon =0\) . By Corollary REF  there are two possibilities:
in case \((i)\)  we have \(\gamma = m =3\)  so the curve \(X\)  is trigonal.
Otherwise, in case \((ii)\)  of Corollary REF , we have that \(\gamma = m+1 =4\)  so \(X\)  is fourgonal.
\((iii)\)  In this case
\(\frac{d-1}{r-1} = \frac{3r-2}{r-1}\) 
hence \(m=3\) . Therefore \(d = 3(r-1) + \epsilon +1 = 3r-1\)  and, so, \(\epsilon =1\) . So the situation is described by
Corollary REF  - (ii) and, in particular, \(X\)  possesses a \(g^1_4\) .
Concerning the slope inequalities, the cases described above behave as follows: as far as \(X\)  is trigonal, all the slope inequalities are fulfilled (see Remark REF ). The case \(d=3r-1\)  is described by Lange–Martens (see [3]}) as follows.
Proposition 5.2 
For any \(r \ge 2\)  and any extremal curve \(X\)  of degree
\(d=3r-1\)  in \(\mathbb {P}^r\) , we have
\(\frac{d_r}{r} < \frac{d_{r+1} }{r+1}.\) 

Finally, the case \(d=3r-2\)  is studied in the next Theorem REF . Its proof will involve the following two results (see, respectively, [3]}).
Proposition 5.3 
Let \(g_\delta ^s\)  be a very ample linear series on \(X\)  with \(\delta \ge 3s - 1\)  and \(\epsilon _{\delta ,s} \ne 0\) . If
\(g > \pi (\delta , s) - m+ 2\) , then \( 2\delta \le g + 3s - 2\) .

Lemma 5.4 
Let \(X\)  be a curve admitting a \(g_\delta ^s\)  with \(\delta \ge 2s-1 \ge 3\)  such that

\(d_{s-1} =\delta -1\) , and

\( 2\delta \le g + 3s - 2\) .

Then \(d_s = \delta \)  and the linear series \(g_\delta ^s\)  is complete and very ample.
Moreover, if \(g_{\delta ^{\prime }}^{s^{\prime }} = |K_X - g_\delta ^s|\)  (hence \(\delta ^{\prime } = 2g-2-\delta \)  and \(s^{\prime } = g-1-\delta +s\) ), we have
\(s^{\prime } \ge s\) ,
\(d_{s^{\prime } +1} \ge \delta ^{\prime }+3\)  and, so,
\(\frac{d_{s^{\prime }} }{s^{\prime }} < \frac{d_{{s^{\prime }} +1} }{{s^{\prime }} +1}.\) 

By Proposition REF  extremal curves in \(\mathbb {P}^r\)  of degree \(d=3r-2\)  can be either trigonal or fourgonal. Since we know that trigonal curves satisfy all slope inequalities, we shall focus the fourgonal case.
We shall see that for \(r=4\)  extremal curves of degree \(3r-2=10\)  satisfy all slope inequalities, while for \(r \ge 5\) , such curves violate the \(r\) -th slope inequality.
Proposition 5.5 
Let \(X \subset \mathbb {P}^4\)  be a fourgonal extremal curve of degree 10. Then \(X\)  satisfies all slope inequalities.

Since \(X\)  is extremal, we have \(g(X)= \pi (10,4)=9\) .
The curves of genus \(g \le 13\)  violating some slope inequality have been classified in [4]}.
In particular, in genus \(g=9\)  the only examples are the extremal curves of degree 8 in \(\mathbb {P}^3\) . The gonality sequence of such curves has been determined in [3]}, and in particular it satisfies \(d_4=11\) . But in our case \(X\)  possesses a \(g^4_{10}\) , so \(d_4 \le 10\) . It follows that no slope inequality is violated.
Now we turn to the case \(r\ge 5\) .
Theorem 5.6 
Let \(X\subset \mathbb {P}^r\)  with \(r \ge 5\)  be a fourgonal extremal curve of degree
\(d= 3r-2.\) 
Then \(d_{r+1} = 3r +1\) .
In particular, \(X\)  violates the \(r\) -th slope inequality:
\(\frac{d_r }{ r} < \frac{d_{r+1}}{ {r+1}}.\) 

We note, first, that \(X\)  is not isomorphic to a plane curve; indeed, in this case we would have \(r=5\)  and \(d=3r-2=13\) , while such curves lie on the Veronese surface and hence their degree is even.
Therefore \(X\)  lies on a rational normal surface.
As observed in Proposition REF , \(m_{d,r}=3\) , \(\epsilon _{d,r} =0\) . Therefore by Theorem REF  and Corollary REF , the class of \(X\)  on a ruled surface \(R \subset \mathbb {P}^r\)  (of degree \(r-1\) ) is given by
\(X \sim 4H -(r-2)L\) 
and
the genus \(g\)  of \(X\)  turns out to be
\(g= \pi (3r-2,r)= 3(r-1).\) 
It is not difficult to
verify that the proof of Theorem REF -\((i)\)  applies also to our case,
and since \(\gamma = 4\) , we have
\(d_{r+1} \le d+3 = 3r+1.\) 
To prove that \(d_{r+1} \ge 3r+1\)  for \(r\ge 5\) , we claim that \(X\)  admits also an embedding in \(\mathbb {P}^{r-2}\)  as an extremal curve.
Keeping the notation of Remark REF , if \(H \sim C_0 + \beta L\) , then it is straightforward to see that \(\beta \ge -C_0^2\)  and the equality holds if and only if \(R\)  is a cone (see, for instance, [9]}).
Since \(X\)  is irreducible we necessarily have
\(0\le X \cdot C_0 = (4C_0 +(4\beta -r+2)L)\cdot C_0 = 4C_0^2 + 4\beta -r+2,\) 
so that
\(\beta \ge -C_0^2+ \frac{r-2 }{ 4}> -C_0^2,\) 
hence
\(\beta \ge -C_0^2 +1,\) 
in particular \(R\)  is not a cone. Therefore we can consider the projection of \(R\)  from a line \(L\)  of its ruling. Hence
consider
the divisor \(H^{\prime }:= H -L\)  on \(R\) ;
taking into account (REF )
we see that the linear system \(|H^{\prime }|\) 
maps \(R\)  in \(\mathbb {P}^{r-2}\)  as a degree \(r-3\)  rational normal surface or a degree \(r-3\)  rational normal cone
(see, for instance, [9]}). Under such a morphism the image of \(X\)  has degree
\(X \cdot H^{\prime }= (4H -(r-2)L) \cdot (H-L)=4H^2 -(r-2)-4=4(r-1)-r-2=3r-6.\) 
As \(m_{3r-6,r-2}=\left[ \frac{3r-7 }{ r-3} \right] =3\)  and \(\epsilon _{3r-6,r-2}=2\) , the maximal genus is in this case
\(\pi (3r-6,r-2)=3((r-3)+2)= 3(r-1)=g(X),\) 
which proves the claim.
The above construction provides a divisor \(H^{\prime } \in g^s_\delta := g^{r-2}_{d-4}\) , so we can consider the birational morphism
\(\varphi _{H^{\prime }}: \; X \rightarrow Y:= \varphi _{H^{\prime }}(X) \subset \mathbb {P}^s\) 
where \(Y\)  is an extremal curve of genus \(g = g(X)= 3r-3\) ,
\(\deg (Y) = \delta = d-4= 3r-6 \quad \hbox{and} \quad s = r-2.\) 
Consequentely, \(Y \subset \mathbb {P}^s\)  is smooth and \(g^s_\delta \)  is very ample. Moreover, since \(r \ge 5\)  then \(s \ge 3\) .
Furthermore, since \(Y\)  is an extremal curve, \(\delta \ge 3s -1\)  and \(\epsilon _{\delta ,s} =2\) , we can apply both Theorem REF  and
Proposition REF , obtaining, respectively, that
\(d_{s-1} = \delta -1\)  and
\(2 \delta \le g +3s -2\) .
Therefore all the assumptions of
Lemma REF  are verified and from it we obtain that \(d_s = \delta \)  and the Serre dual series \(g_{\delta ^{\prime }}^{s^{\prime }} = |K_X - g_\delta ^s|\)  of \(g^s_\delta \)  satisfies the following relation
\(d_{s^{\prime } +1} \ge \delta ^{\prime }+3,\) 
where
\(\delta ^{\prime } = 2g-2-\delta = 2(3r-3)-2 - (3r-6)=3r-2,\) 
\(s^{\prime } = g-1-\delta +s = 3r-3-1 -(3r-6) + r-2=r.\) 
Therefore the above inequality gives
\(d_{r +1} \ge 3r+1,\) 
as required.
Finally, since \(d_r \le d\) , we have
\(\frac{d_r}{r}\le \frac{d}{r} = \frac{3r-2}{r} < \frac{3r+1}{r+1}= \frac{d_{r+1}}{r+1}.\) 

Fourgonal extremal curves
The setting of the current section is the following: \(X\)  is a fourgonal extremal curve in \(\mathbb {P}^r\) 
(where \(r \ge 3\) ) of degree \(d\) , genus \(g\)  and \(m\) -ratio \(m\) .
Remark 6.1 Concerning space curves (i.e. the case \(r=3\) ), let us consider the assumptions (a) and (b) of
Theorem REF -\((ii)\) .
Since \(0 \le \epsilon \le r-2=1\) , then
the only possibility is case (a), where \(\epsilon =0\)  and \(m=\gamma \le r+1 = 4\) . For this reason, regarding space curves, Theorem REF  describes only the fourgonal case. More precisely, if \(X\)  is a fourgonal extremal curve of degree \(d\)  in \(\mathbb {P}^3\)  then it satisfies the following conditions:
(i)
\(d_3 = d\) ;
(ii)
\(d_4\le d+3\) .
(iii)
If, in addition, we assume \(\epsilon = 0\)  and \(m=4\) , then the 3rd slope inequality holds, i.e.
\(\frac{d_3}{3} \ge \frac{d_4 }{4}.\) 

Note that this situation is quite specific. Namely, formulas (REF ) and (REF ) give \(d=9\)  and \(g=12\) .
So, comparing this fact with (REF ) (i.e. the violation of the third slope inequality), which holds for \(d \ge 10\) , we obtain that this bound on the degree given by Kato–Martens is sharp. Moreover, the above claims fit with the description of the gonal sequence given by the same authors in [4]}.

It is easy to specialize the results of Section 4 to the following results in the fourgonal case.
Corollary 6.2 
Let \(X\)  be a fourgonal extremal curve of degree \(d \ge 3r-1\)  in \(\mathbb {P}^r\) , where \(r \ge 3\) .
Then \(X\)  satisfies the following conditions:
\(i)\)
\(d_r = d\) ;
\(ii)\)
\(d_{r+1}\le d+3\) .
\(iii)\)
If, in addition, we assume that \(d \ge 3r\)  , then the \(r\) -th slope inequality holds, i.e.
\(\frac{d_r}{r} \ge \frac{d_{r+1} }{r+1}.\) 


Note first that \(X\)  cannot be isomorphic to a plane curve (otherwise we would have \(r=5\)  and \(d =2k \ge 14\) , so its gonality would be \(k-1\ge 6\) ). Therefore we can apply Theorem REF  and obtain \((i)\)  and \((ii)\) . Morever, \(d \ge 3r\)  and \(\gamma =4\)  imply that \(\epsilon \ne 1\)  (see Table 1). Hence either assumption \((a)\)  or \((b)\)  in Theorem REF  hold.
Moreover, a stronger form of Corollary REF  holds.
Corollary 6.3 Let \(X\)  be an extremal curve of degree \(d\)  in \(\mathbb {P}^r\) , where \(r \ge 3\) .
Then the following conditions are equivalent:
\(i)\)
the degree satisfies \(3r \le d \le 4r-4\) ;
\(ii)\)
\(X\)  is fourgonal and the \(r\) -th slope inequality holds.


Immediate, from Table 1.
As a consequence of the results of the previous sections, we are able to determine the gonal sequence of some extremal curve in a certain interval and to show that all the slope inequalities hold there. Indeed, on one hand, in Theorem REF  we have seen that, by projecting an extremal curve lying in \(\mathbb {P}^r\)  from a gonal divisor, we may obtain an extremal curve in \(\mathbb {P}^{r-2}\) .
On the other hand, for an extremal curve in \(\mathbb {P}^r\) , the values of \(d_{r}\)  and \(d_{r-1}\)  are determined by Theorems REF  and REF .
Theorem 6.4 
Let
\(X \sim 4(C_0 +n L) \subset \mathbb {F}_n\) 
be a smooth irreducible curve, where \(n \ge 3\) .
For any integer \(a\)  such that
\(0\le a \le \left[ \frac{n-3}{ 2}\right],\) 
the gonal sequence of \(X\)  satisfies
\(d_{n+2a} = 4(n+a)-1, \quad d_{n+2a+1}= 4(n+a).\) 
Moreover, the following bound holds
\(d_{n+2\left[\frac{n-3}{ 2}\right]+2} \le 4n+4\left[\frac{n-3}{ 2}\right]+3.\) 
Consequentely, for any \(n\le r \le n+2\left[ \frac{n-3}{ 2}\right]+1\) , the \(r\) -th slope inequality holds.

For any integer \(a\)  as above, consider the divisor \(H_a\)  on \(\mathbb {F}_n\)  defined by
\(H_a=C_0 +(n+a)L\) 
and set
\(r_a:=n+2a+1.\) 
By Proposition REF -(a) and (REF ), if \(a \ne 0\) , then the linear system \(|H_a|\)  embeds \(\mathbb {F}_n\)  isomorphically in \({\mathbb {P}}^{r_a}\) .
If \(a=0\) , then \(\varphi _{H_0} (\mathbb {F}_n)\)  is a ruled surface of degree \(H_0^2 =n\)  in \(\mathbb {P}^{n+1}\) . Moreover
\(\deg (\varphi _{H_0}(C_0)) = C_0 \cdot H_0 = 0\) 
so the unisecant \(C_0\)  is contracted to a point. Hence \(\varphi _{H_0} (\mathbb {F}_n)\)  is a rational normal cone and
\(\varphi _{H_0}\)  is an isomorphism between
\(\mathbb {F}_n \setminus C_0\)  and its image.
In both cases, the corresponding morphism \(\varphi _{H_a}\)  is an isomorphism on \(X\) . Namely, if \(a \ne 0\)  it is clear. If \(a = 0\) , it follows from \(X \cdot C_0 = 4(C_0 + nL) \cdot C_0 =0\) , so \(X \subset (\mathbb {F}_n \setminus C_0)\) .
Therefore, for any \(a \ge 0\) , the curve \(X_a:= \varphi _{H_a}(X) \subset \mathbb {P}^{r_a}\)  has degree
\(\delta _a :=X \cdot H_a = 4(C_0 +nL)\cdot (C_0 +(n+a)L)=4(n+a).\) 
We claim that \(X_a\)  is an extremal curve in \(\mathbb {P}^{r_a}\) . In order to show this, we compute first the genus of \(X\) . As usual, the Adjunction formula gives
\(\begin{array}{ll}2g-2& =(K_{\mathbb {F}_n} +X) \cdot X =(-2C_0 -(2+n)L + 4 C_0+4n L) \cdot (4 C_0+4n L)=\\&= 12 n -8.\\\end{array}\) 
Therefore we obtain
\(g=6n-3.\) 
Now we compute the Castelnuovo bound of the genus.
Taking into account the bound (REF ) of the integer \(a\) , we obtain that
\(m_{\delta _a,r_a}=\left[ \frac{4n+4a-1}{ n+2a}\right]=3, \quad \epsilon _{\delta _a,r_a}=n-2a-1\) 
and
\(2 \le \epsilon _{\delta _a, r_a} \le n-1.\) 
An immediate computation of the Castelnuovo bound given in (REF )
shows that
\(\pi (\delta _a, r_a)= 3(n+2a+\epsilon _{\delta _a,r_a})=3(2n-1)=6n-3,\) 
which coincides with the value of \(g\)  determined in (REF ), so \(X_a \subset \mathbb {P}^{r_a}\)  is an extremal curve.
In order to prove (REF ), note first that
\(\epsilon _{\delta _a,r_a} \ge 2\)  implies that \(n-2a \ge 3\) ; therefore \(\delta _a \ge 3r_a \) . So we can apply the cited result of Lange–Martens (see Theorem REF ) to the extremal curve \(X_a \subset \mathbb {P}^{r_a}\) , obtaining that
\(d_{n+2a}=\delta _a -1 = 4(n+a)-1.\) 
Second, since \(X_a\)  is a fourgonal extremal curve and \(r_a \ge 3\)  (again from (REF )), we can apply also Corollary REF -\((i)\) , obtaining that
\(d_{n+2a+1}=\delta _a = 4(n+a).\) 
Therefore (REF ) is proved. As a consequence, for any \(0\le a \le \left[ \frac{n-3}{ 2}\right]\) ,
both the \((r_a-1)\) -th and \(r_a\) -th slope inequalities are satisfied.
Now consider the highest value \(\overline{a}=\left[\frac{n-3}{ 2}\right]\) . Also in this case \(X_{\overline{a}}\)  is an extremal fourgonal curve of degree \(\delta _{\overline{a}}\)  in
\(\mathbb {P}^{r_{\overline{a}}}\) . Hence, by
Corollary REF -\((ii)\) ,
\(d_{n+2{\overline{a}}+2} \le 4(n+{\overline{a}})+3\) 
hence also (REF ) is proved.
Finally, as shown before, \(\delta _a \ge 3r_a \)  and so Corollary REF -\((iii)\)  yields the \(r\) -th slope inequality in the considered range of \(r\) .
Remark 6.5 The gonal subsequence (REF ) and the bound (REF )
in Theorem REF  can be explicitly written in both the following cases, according to the parity of \(n\) .
If \(n\)  is even, i.e. \(n = 2k\)  for some \(k\) , then \(2\left[ \frac{n-3}{ 2}\right] =2(k-2) = n-4\)  so
\(d_{n}= 4n-1, \ d_{n+1}= 4n, \ d_{n+2}= 4n+3, \ d_{n+3} = 4n +4, \ \dots ,\ d_{2n-4}= 6n-9, \ d_{2n-3}=6n-8\) 
and \(d_{2n-2}\le 6n-5\) .
If \(n\)  is odd, i.e. \(n = 2k+1\)  for some \(k\) , then \(2\left[ \frac{n-3}{ 2}\right] =2(k-1) = n-3\)  so
\(d_{n}= 4n-1, \ d_{n+1}= 4n, \ d_{n+2}= 4n+3, \ d_{n+3} = 4n +4, \ \dots ,\ d_{2n-3}= 6n-7, \ d_{2n-2}=6n-6\) 
and \(d_{2n-1}\le 6n-3\) .

Theorem REF  shows that, as far as \(n \ge 5\) , there are classes of extremal curves in \(\mathbb {P}^n\)  which gonal sequence does not follow the pattern given in [4]}, for extremal curves in \(\mathbb {P}^3\) .
Moreover, using the results in [13]} and [14]}, it is possible to verify that the curves of Theorem REF  are very special curves in the fourgonal moduli space. Therefore the problem of describing the locus of curves violating the slope inequalities is intriguing and deserves further investigations.
